How to describe arbitrary directed graph if I have a set?

Question

I have an arbitrary set $A$ which is a subset of another set $A \subset B$, the elements of the set $A$ constitue a directed graph. If I want to describe it via notation (algebra) without mentioning the concrete elements, how can I do that? Is that even possible?

Answer 1

There is no notation for the statement "something is a directed graph". This is probably because it's hard to imagine a situation where you have an object, but you're not sure if it's a directed graph or not. (There is no larger class of objects, some of which are directed graphs and some of which are not, such that you want to say "that object in the larger class? it's actually a directed graph".)

You can use notation for the other relationship in your problem. For example, if you have a directed graph $D$, you could write $V(D) \subseteq B$ to indicate that the vertices of $D$ are a subset of $B$, or $E(D) \subseteq B$ to indicate that the edges of $D$ are a subset of $B$.